Bayesian classification methods

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1 Byes lssfto methods

2 Outle Bkgroud robblty Bss robblst Clssfto Nïve Byes rple d Algorthms Emple: ly Tes Zero Codtol robblty Summry

3 Bkgroud There re three methods to estblsh lssfer Model lssfto rule dretly k-nerest Neghbor Deso trees Neurl etworks Support Vetor Mhes b Mke probblst model of dt wth eh lss Byes Rule Nve Byes Byes Networks Geertve models 3

4 robblty Bss ror odtol d jot probblty for rdom vrbles ror probblty: Codtol probblty: Jot probblty: Reltoshp: Idepedee: Byes Rule Dsrmtve Geertve Lkelhood ror osteror Evdee 4

5 robblty Bss Quz: We hve two s-sded de. Whe they re tolled t ould ed up wth the followg oure: A de lds o sde 3 B de lds o sde d C Two de sum to eght. Aswer the followg questos: A 3 B C??? 4 A B 5 C A 6 A B 7 A C 8 Is A C equl to???? A C? 5

6 robblst Clssfto Estblshg probblst model for lssfto Dsrmtve model L Dsrmtve robblst Clssfer L To tr dsrmtve lssfer regrdless ts probblst or oprobblst ture ll trg emples of dfferet lsses must be jotly used to buld up sgle dsrmtve lssfer. Output L probbltes for L lss lbels probblst lssfer whle sgle lbel s heved by o-probblst lssfer. 6

7 robblst Clssfto Estblshg probblst model for lssfto ot. Geertve model must be probblst L Geertve robblst Model for Clss Geertve robblst Model for Clss L L L probblst models hve to be tred depedetly Eh s tred o oly the emples of the sme lbel Output L probbltes for gve put wth L models Geertve mes tht suh model produes dt subjet to the dstrbuto v smplg. 7

8 robblst Clssfto Mmum A osteror MA lssfto rule For put fd the lrgest oe from L probbltes output by dsrmtve probblst lssfer... L. Assg to lbel * f * s the lrgest. Geertve lssfto wth the MA rule Apply Byes rule to overt them to posteror probbltes for The pply the MA rule to ssg lbel L Commo ftor for ll L probbltes 8

9 9 Nïve Byes Byes lssfto Dffulty: lerg the jot probblty s fesble! Nïve Byes lssfto Assume ll put fetures re lss odtolly depedet! Apply the MA lssfto rule: ssg to * f.... for L L ] [ ] [ * * * * > ' Applyg the depedee ssumpto of estmte * of estmte

10 Nïve Byes For eh trget vlue of ˆ estmte For every feture vlue ˆ j jk wth emples S; jk estmte of eh feture jk L j j F; k wth emples S; N j 'ʹ ʹ ʹ ˆ ʹ ˆ ʹ ˆ ʹ * * * * [ ]ˆ > [ ]ˆ ˆ ʹ L 0

11 Emple Emple: ly Tes

12 Emple Lerg hse Outlook lyyes lyno Suy /9 3/5 Overst 4/9 0/5 R 3/9 /5 Temperture lyyes lyno Hot /9 /5 Mld 4/9 /5 Cool 3/9 /5 Humdty lyyes lyno Wd lyyes lyno Hgh 3/9 4/5 Norml 6/9 /5 Strog 3/9 3/5 Wek 6/9 /5 lyyes 9/4 lyno 5/4

13 Emple Test hse Gve ew ste predt ts lbel OutlookSuy TempertureCool HumdtyHgh WdStrog Look up tbles heved the lerg phrse OutlookSuylyYes /9 TempertureCoollyYes 3/9 HumtyHghlyYes 3/9 WdStroglyYes 3/9 lyyes 9/4 Deso mkg wth the MA rule OutlookSuylyNo 3/5 TempertureCoollyNo /5 HumtyHghlyNo 4/5 WdStroglyNo 3/5 lyno 5/4 Yes [SuyYesCoolYesHghYesStrogYes]lyYes No [SuyNo CoolNoHghNoStrogNo]lyNo Gve the ft Yes < No we lbel to be No. 3

14 Nïve Byes Algorthm: Cotuous-vlued Fetures Numberless vlues tke by otuous-vlued feture Codtol probblty ofte modeled wth the orml dstrbuto µ σ j j j µ j ˆ j ep πσ j σ j : me vergeof feture vlues of emples : stdrd devto of feturevlues Lerg hse: for X X X C L Output: L orml dstrbutos d C L Test hse: Gve ukow ste Xʹ ʹ ʹ Isted of lookg-up tbles lulte odtol probbltes wth ll the orml dstrbutos heved the lerg phrse Apply the MA rule to ssg lbel the sme s doe for the dsrete se j j of emples for whh for whh 4

15 Nïve Byes Emple: Cotuous-vlued Fetures Temperture s turlly of otuous vlue. Yes: No: Estmte me d vre for eh lss N N µ Yes.64 µ σ µ N µ No 3.88 N Lerg hse: output two Guss models for tempc ˆ Yes.35 ˆ No 7.09 ep π ep π σ σ Yes No ep π ep π

16 Zero odtol probblty If o emple ots the feture vlue I ths rumste we fe zero odtol probblty problem durg test ˆ ˆ jk ˆ 0 for ˆ jk jk For remedy lss odtol probbltes re-estmted wth :umber of :umber of p + mp ˆ jk + m trg emples for whh trg emples for whh :pror estmte usully m : weght to pror umber of p / t j for m-estmte t "vrtul"emples j jk 0 d possble vlues of m j 6

17 Zero odtol probblty Emple: outlookoversto0 the ply-tes dtset Addg m vrtul emples m: up to % of #trg emple I ths dtset # of trg emples for the o lss s 5. We oly dd m vrtul emple our m-estmte remedy. The outlook feture tkes oly 3 vlues. So p/3. Re-estmte outlooko wth the m-estmte 7

18 Summry Nïve Byes: the odtol depedee ssumpto Trg d test re very effet Two dfferet dt types led to two dfferet lerg lgorthms Workg well sometmes for dt voltg the ssumpto! A populr geertve model erforme ompettve to most of stte-of-the-rt lssfers eve presee of voltg depedee ssumpto My suessful ppltos e.g. spm ml flterg A good ddte of bse lerer esemble lerg Aprt from lssfto ïve Byes do more 8

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